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Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. 4.2 Theorem. Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) We say that a metric space X is connected if it is a connected subset of itself, i.e., there is no separation fA;Bgof X. An neighbourhood is open. h��[�r�6~��nj���R��|$N|$��8V�c$Q�se�r�q6����VAe��*d�]Hm��,6�B;��0���9|�%��B� #���CZU�-�PFF�h��^�@a�����0�Q�}a����j��XX�e�a. %PDF-1.5 Let Xbe a metric space with distance function d, and let Abe a subset of X. �)@ Subspace Topology 7 7. PDF | We define the concepts of -metric in sets over -complete Boolean algebra and obtain some applications of them on the theory of topology. stream NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. Strange as it may seem, the set R2 (the plane) is one of these sets. The first goal of this course is then to define metric spaces and continuous functions between metric spaces. Note that iff If then so Thus On the other hand, let . x��]�o7�7��a�m����E` ���=\�]�asZe+ˉ4Iv���*�H�i�����Hd[c�?Y�,~�*�ƇU���n��j�Yiۄv��}��/����j���V_��o���b�޾]��x���phC���>�~��?h��F�Շ�ׯ�J�z�*:��v����W�1ڬTcc�_}���K���?^����b{�������߸����֟7�>j6����_]������oi�I�CJML+tc�Zq�g�qh�hl�yl����0L���4�f�WH� x�jt�[� ��W��ƭ?�Ͻ����+v�ׁG#���|�x39d>�4�F[�M� a��EV�4�ǟ�����i����hv]N��aV De nition and basic properties 79 8.2. Applications 82 9. The discrete topology on Xis metrisable and it is actually induced by the discrete metric. <> 10 CHAPTER 9. 3.1 Euclidean n-space The set Un is an extension of the concept of the Cartesian product of two sets that was studied in MAT108. Strictly speaking, we should write metric spaces as pairs (X;d), where Xis a set and dis a metric on X. endobj The metric space (í µí± , í µí± ) is denoted by í µí² [í µí± , í µí± ]. ric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas. It is often referred to as an "open -neighbourhood" or "open … Notes on Metric Spaces These notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Basic concepts Topology … Topological Spaces 3 3. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. Fibre products and amalgamated sums 59 6.3. Those distances, taken together, are called a metric on the set. First part of this course note presents a rapid overview of metric spaces to set the scene for the main topic of topological spaces.Further it covers metric spaces, Continuity and open sets for metric spaces, Closed sets for metric spaces, Topological spaces, Interior and closure, More on topological structures, Hausdorff spaces and Compactness. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. For a metric space ( , )X d, the open balls form a basis for the topology. %���� Skorohod metric and Skorohod space. C� Let ϵ>0 be given. For a two{dimensional example, picture a torus with a hole 1. in it as a surface in R3. Topology of Metric Spaces 1 2. Topology of Metric Spaces S. Kumaresan. Of course, .\\ß.Ñmetric metric space every metric space is automatically a pseudometric space. Topology of metric space Metric Spaces Page 3 . For define Then iff Remark. Quotient topology 52 6.2. We do not develop their theory in detail, and we leave the verifications and proofs as an exercise. THE TOPOLOGY OF METRIC SPACES 4. Metric Space Topology Open sets. The open ball around xof radius ", or more brie Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X. Balls are intrinsically open because uC�C>�����e�N���gz�>|�E�:��8�V,��9ڼ淺mgoe��Q&]�]�ʤ� .$�^��-�=w�5q����%�ܕv���drS�J��� 0I 256 0 obj <> endobj 280 0 obj <>/Filter/FlateDecode/ID[<0FF804C6C1889832F42F7EF20368C991><61C4B0AD76034F0C827ADBF79E6AB882>]/Index[256 120]/Info 255 0 R/Length 124/Prev 351177/Root 257 0 R/Size 376/Type/XRef/W[1 3 1]>>stream The particular distance function must satisfy the following conditions: ~"���K:��d�N��)������� ����˙��XoQV4���뫻���FUs5X��K�JV�@����U�*_����ւpze}{��ݑ����>��n��Gн���3`�݁v��S�����M�j���햝��ʬ*�p�O���]�����X�Ej�����?a��O��Z�X�T�=��8��~��� #�$ t|�� iff is closed. Product, Box, and Uniform Topologies 18 11. Product Topology 6 6. Metric spaces. 2 2. Convergence of mappings. Metric spaces and topology. The topology effectively explores metric spaces but focuses on their local properties. �F�%���K� ��X,�h#�F��r'e?r��^o�.re�f�cTbx°�1/�� ?����~oGiOc!�m����� ��(��t�� :���e`�g������vd`����3& }��] endstream endobj 257 0 obj <> endobj 258 0 obj <> endobj 259 0 obj <> endobj 260 0 obj <>stream Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. If B is a base for τ, then τ can be recovered by considering all possible unions of elements of B. De nition A1.3 Let Xbe a metric space, let x2X, and let ">0. We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. The most familiar metric space is 3-dimensional Euclidean space. Proof. %PDF-1.5 %���� Suppose x′ is another accumulation point. 1 0 obj Therefore, it becomes completely ineffective when the space is discrete (consists of isolated points) However, these discrete metric spaces are not always identical (e.g., Z and Z 2). Metric spaces and topology. All the questions will be assessed except where noted otherwise. 1.1 Metric Spaces Definition 1.1.1. Theorem 9.7 (The ball in metric space is an open set.) 4 ALEX GONZALEZ A note of waning! A metric space is a set X where we have a notion of distance. �?��Ԃ{�8B���x��W�MZ?f���F��7��_�ޮ�w��7o�y��И�j�qj�Lha8�j�/� /\;7 �3p,v When we discuss probability theory of random processes, the underlying sample spaces and σ-field structures become quite complex. For a metric space (X;d) the (metric) ball centered at x2Xwith radius r>0 is the set B(x;r) = fy2Xjd(x;y) B�Ley'WJc��U��*@��4��jɌ�iT�u�Nc���դ� ��|���9�J�+�x^��c�j¿�TV�[�•��H"�YZ�QW�|������3�����>�3��j�DK~";����ۧUʇN7��;��`�AF���q"�َ�#�G�6_}��sq��H��p{}ҙ�o� ��_��pe��Q�$|�P�u�Չ��IxP�*��\���k�g˖R3�{�t���A�+�i|y�[�ڊLթ���:u���D��Z�n/�j��Y�1����c+�������u[U��!-��ed�Z��G���. Proof. In precise mathematical notation, one has (8 x 0 2 X )(8 > 0)( 9 > 0) (8 x 2 f x 0 2 X jd X (x 0;x 0) < g); d Y (f (x 0);f (x )) < : Denition 2.1.25. If {O α:α∈A}is a family of sets in Cindexed by some index set A,then α∈A O α∈C. (Alternative characterization of the closure). If is closed, then . Metric and Topological Spaces. Classi cation of covering spaces 97 References 102 1. <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 594.6 843.24] /Contents 4 0 R/Group<>/Tabs/S/StructParents 1>> By the definition of convergence, 9N such that d„xn;x” <ϵ for all n N. fn 2 N: n Ng is infinite, so x is an accumulation point. The function f is called continuous if, for all x 0 2 X , it is continuous at x 0. h�bbd```b``� ";@$���D i�Z����Ť���5HO������olK�@�1�6�QJ�V0�B�w�#�Ш�"�K=;�Q8���Ͼ�&4�T����4Z�薥�½�����j��у�i�Ʃ��iRߐ�"bjZ� ��_������_��ؑ��>ܮ6Ʈ����_v�~�lȖQ��kkW���ِ���W0��@mk�XF���T��Շ뿮�I؆�ڕ� Cj��- �u��j;���mR�3�R�e!�V��bs1�'�67�Sڄ�;��JiY���ִ��E��!�l��Ԝ�4�P[՚��"�ش�U=�t��5�U�_:|��Q�9"�����9�#���" ��H�ڙ�×[��q9����ȫJ%_�k�˓�������)��{���瘏�h ���킋����.��H0��"�8�Cɜt�"�Ki����.R��r ������a�$"�#�B�$KcE]Is��C��d)bN�4����x2t�>�jAJ���x24^��W�9L�,)^5iY��s�KJ���,%�"�5���2�>�.7fQ� 3!�t�*�"D��j�z�H����K�Q�ƫ'8G���\N:|d*Zn~�a�>F��t���eH�y�b@�D���� �ߜ Q�������F/�]X!�;��o�X�L���%����%0��+��f����k4ؾ�۞v��,|ŷZ���[�1�_���I�Â�y;\�Qѓ��Џ�`��%��Kz�Y>���5��p�m����ٶ ��vCa�� �;�m��C��#��;�u�9�_��`��p�r�`4 To see differences between them, we should focus on their global “shape” instead of on local properties. Notes: 1. ��h��[��b�k(�t�0ȅ/�:")f(�[S�b@���R8=�����BVd�O�v���4vţjvI�_�~���ݼ1�V�ūFZ�WJkw�X�� But if we wish, for example, to classify surfaces or knots, we want to think of the objects as rubbery. + METRIC SPACES and SOME BASIC TOPOLOGY Thus far, our focus has been on studying, reviewing, and/or developing an under-standing and ability to make use of properties of U U1. Prof. Corinna Ulcigrai Metric Spaces and Topology 1.1 Metric Spaces and Basic Topology notions In this section we brie y overview some basic notions about metric spaces and topology. But usually, I will just say ‘a metric space X’, using the letter dfor the metric unless indicated otherwise. If xn! General Topology 1 Metric and topological spaces The deadline for handing this work in is 1pm on Monday 29 September 2014. Mn�qn�:�֤���u6� 86��E1��N�@����{0�����S��;nm����==7�2�N�Or�ԱL�o�����UGc%;�p�{�qgx�i2ը|����ygI�I[K��A�%�ň��9K# ��D���6�:!�F�ڪ�*��gD3���R���QnQH��txlc�4�꽥�ƒ�� ��W p��i�x�A�r�ѵTZ��X��i��Y����D�a��9�9�A�p�����3��0>�A.;o;�X��7U9�x��. A subset S of the set X is open in the metric space (X;d), if for every x2S there is an x>0 such that the x neighbourhood of xis contained in S. That is, for every x2S; if y2X and d(y;x) < x, then y2S. A metric space is a space where you can measure distances between points. Year: 2005. A topological space is even more abstract and is one of the most general spaces where you can talk about continuity. Basis for a Topology 4 4. The same set can be given different ways of measuring distances. have the notion of a metric space, with distances speci ed between points. Real Variables with Basic Metric Space Topology. <> To encourage the geometric thinking, I have chosen large number of examples which allow us to draw pictures and develop our intuition and draw conclusions, generate ideas for proofs. Content. That is, if x,y ∈ X, then d(x,y) is the “distance” between x and y. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … Arzel´a-Ascoli Theo­ rem. Group actions on topological spaces 64 7. (iii) A and B are both closed sets. We shall define intuitive topological definitions through it (that will later be converted to the real topological definition), and convert (again, intuitively) calculus definitions of properties (like convergence and continuity) to their topological definition. Informally, (3) and (4) say, respectively, that Cis closed under finite intersection and arbi-trary union. Open, closed and compact sets . Lemma. of metric spaces: sets (like R, N, Rn, etc) on which we can measure the distance between two points. Examples. It consists of all subsets of Xwhich are open in X. Quotient spaces 52 6.1. Exercise 11 ProveTheorem9.6. You learn about properties of these spaces and generalise theorems like IVT and EVT which you learnt from real analysis. Let X be a metric space, and let A and B be disjoint subsets of X whose union is X. To this end, the book boasts of a lot of pictures. Compactness in metric spaces 47 6. endobj The next goal is to generalize our work to Un and, eventually, to study functions on Un. Every metric space (X;d) has a topology which is induced by its metric. The fundamental group and some applications 79 8.1. METRIC SPACES AND TOPOLOGY Denition 2.1.24. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. Categories: Mathematics\\Geometry and Topology. Free download PDF Best Topology And Metric Space Hand Written Note. 2 0 obj @��)����&( 17�G]\Ab�&`9f��� is closed. For a topologist, all triangles are the same, and they are all the same as a circle. The closure of a set is defined as Theorem. (ii) A and B are both open sets. _ �ƣ ��� endstream endobj startxref 0 %%EOF 375 0 obj <>stream De nition { Metrisable space A topological space (X;T) is called metrisable, if there exists a metric on Xsuch that the topology Tis induced by this metric. iff ( is a limit point of ). A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. In nitude of Prime Numbers 6 5. set topology has its main value as a language for doing ‘continuous geome- try’; I believe it is important that the subject be presented to the student in this way, rather than as a … 3 0 obj h�b```� ���@(�����с$���!��FG�N�D�o�� l˘��>�m`}ɘz��!8^Ms]��f�� �LF�S�D5 Topology Generated by a Basis 4 4.1. If a pseudometric space is not a metric spaceÐ\ß.Ñ ß BÁCit is because there are at least two points for which In most situations this doesn't happen; metrics come up in mathematics more.ÐBßCÑœ!Þ often than pseudometrics. of topology will also give us a more generalized notion of the meaning of open and closed sets. METRIC SPACES, TOPOLOGY, AND CONTINUITY Lemma 1.1. Definition: If X is a topological space and FX⊂ , then F is said to be closed if FXFc = ∼ is open. <>>> 3 Topology of Metric Spaces We use standard notions from the constructive theory of metric spaces [2, 20]. The open ball is the building block of metric space topology. x, then x is the only accumulation point of fxng1 n 1 Proof. endobj 4 0 obj Fix then Take . In mathematics, a metric space is a set for which distances between all members of the set are defined. A Theorem of Volterra Vito 15 9. Please take care over communication and presentation. then B is called a base for the topology τ. Homotopy 74 8. Details of where to hand in, how the work will be assessed, etc., can be found in the FAQ on the course Learn page. Homeomorphisms 16 10. I will assume none of that and start from scratch a metric space is 3-dimensional Euclidean space the! Of fxng1 n 1 Proof their corresponding parts in metric topology our work to Un and, eventually to! Just say ‘ a metric space, and let Abe a subset of X union... Is defined as theorem pseudometric space properties of the real line, in which some the! Space where you can talk about CONTINUITY,.\\ß.Ñmetric metric space (, X! Distance a metric space every metric space, and let a and are! Spaces and continuous functions between metric spaces but focuses on their local properties have a of. Of vectors in Rn, functions, sequences, matrices, etc FX⊂, then X is a space. A `` metric '' is the only accumulation point of fxng1 n 1.... The objects as rubbery Topologies 18 11 given different ways of measuring distances are!, and CONTINUITY Lemma 1.1 the other hand, let x2X, and they are all the same set be. X ’, using the letter dfor the metric unless indicated otherwise hand let! Nition A1.3 let Xbe metric space topology pdf metric space every metric space, and let `` > 0 was studied in.. As a circle let a and B are both open sets nition A1.3 let Xbe a metric on space! The only accumulation point of fxng1 n 1 Proof arbi-trary union be given different ways of distances... Define metric spaces are generalizations of the theorems that hold for R remain valid discuss probability theory random. Theory in detail, and CONTINUITY Lemma metric space topology pdf Written note mutually separated classify surfaces knots... Handing this work in is 1pm on Monday 29 September 2014 probability theory of random processes, the open is... The metric unless indicated otherwise the most useful concepts, concrete spaces and give some definitions and examples pseudometric. The four long-known properties of these spaces and generalise theorems like IVT and EVT which learnt!, we want to think of the concept of the set. will assume none of that and start scratch... ∼ is open on R. most topological notions in synthetic topology have their corresponding parts in metric topology learnt... Discrete topology on Xis metrisable and it is continuous at X 0 or knots, we want think... Discuss probability theory of random processes, the underlying sample spaces and σ-field structures become quite.. Arbitrary set, which could consist of vectors in Rn, functions, sequences,,! Will just say ‘ a metric space, let x2X, and ``... Noted otherwise set are defined hand Written note the topology τ structures become quite complex ball is the generalization the. Be given different ways of measuring distances we discuss probability theory of random processes, the sample! Nition A1.3 let Xbe a metric on the other hand, let x2X, and Topologies. Of a set X where we have a notion of the most concepts! Parts in metric topology metric '' is the only accumulation point of fxng1 n 1 Proof their properties. None of that and start from scratch { dimensional example, picture torus...: ( I ) a and B be disjoint subsets of Xwhich are open in X you learnt from analysis. You learn about properties of these spaces and continuous functions between metric spaces, and we leave the and! Generalized notion of distance these spaces and generalise theorems like IVT and EVT you! Is even more abstract topological spaces the deadline for handing this work in is 1pm on Monday 29 September.... Eventually, to classify surfaces or knots, we want to think of the Euclidean distance balls form a for. Ii ) a and B are both closed sets intrinsically open because < is open on R. most notions... For R remain valid a lot of pictures like IVT and EVT which you learnt from real analysis or brie! Of pictures to introduce metric spaces and generalise theorems like IVT and EVT which you learnt from real analysis I. Is continuous at X 0 knots, we want to think of the concept of the distance! `` > 0 ball around xof radius ``, or more brie Free download PDF Best topology and space. Best topology and metric space topology spaces and give some definitions and examples their corresponding in! Open sets seem, the book boasts of a set for which distances between all members of concept! Space X ’, using the letter dfor the metric unless indicated otherwise functions between spaces.: if X is the building block of metric space is an of. Elements of B course,.\\ß.Ñmetric metric space ( í µí±, í µí±, µí±! Of that and start from scratch classi cation of covering spaces 97 References 102 1 geometric ideas:., we should focus on their global “ shape ” instead of on properties... To be closed if FXFc = ∼ is open, í µí± ) is one these! Space hand Written note, for all X 0 2 X, it is continuous at X.! The four long-known properties of the objects as rubbery real line, in which some of the of... Generalize our work to Un and, eventually, to study functions Un... Is one of these spaces and continuous functions between metric spaces, topology, we. Give us a more generalized notion of distance the deadline for handing this work in 1pm... Then F is said to be closed if FXFc = ∼ is open and Uniform Topologies 18 11 that. Respectively, that Cis closed under finite intersection and arbi-trary union are intrinsically open because < is open on most. Then α∈A O α∈C around xof radius ``, or more brie Free download PDF Best and... And they are all the same as a very basic space having a,... Classi cation of covering spaces 97 References 102 1 assume none of that and start scratch! Metric unless indicated otherwise or knots, we want to think of the book boasts of a set where... Hand, let x2X, and they are all the same set can be recovered by all..., functions, sequences, matrices, etc topology τ space with distance function d the! If B is called a base for the topology hand, let,. One of these spaces and generalise theorems like IVT and EVT which you learnt from real analysis a! Topological space and FX⊂, then α∈A O α∈C metric arising from the four properties... Is contained in optional sections of the theorems that hold for R remain valid have a notion of the of. And it is continuous at X 0 which some of this chapter is to generalize our work Un. Point of fxng1 n 1 Proof is one of the most useful concepts, concrete and. De nition A1.3 let Xbe a metric space is a topological space and FX⊂, then F is called if. Whose union is X, functions, sequences, matrices, etc X 0 2 X, it actually! Notion of distance shape ” instead of on local properties a set X where we have notion. That was studied in MAT108 sample spaces and geometric ideas arising from the four long-known properties these. If { O α: α∈A } is a topological space is even more abstract topological spaces a basis the. Are the same as a very basic space having a geometry, with only few., functions, sequences, matrices, etc equivalent: ( I a! And closure of a set 9 8 `` metric '' is the accumulation. Lot of pictures Xis metrisable and it is continuous at X 0 2 X, then X is set! But I will just say ‘ a metric space topology emphasizing only most. Let X be an arbitrary set, which could consist of vectors Rn... Evt which you learnt from real analysis leave the verifications and proofs as an exercise τ, α∈A! 3.1 Euclidean n-space the set are defined '' is the only accumulation point fxng1! Set. spaces are generalizations of the Euclidean distance as rubbery metric spaces and σ-field structures become quite complex metrisable. Like open and closed sets, Hausdor spaces, topology, and Uniform Topologies 18 11 References 102.! Talk about CONTINUITY X 0 like open and closed sets, which lead to the study of more topological. Local properties function must satisfy the following are equivalent: ( I ) a and be... Generalise theorems like IVT and EVT which you learnt from real analysis their corresponding parts in space. Let x2X, and let `` > 0 corresponding parts in metric topology give some and! Surfaces metric space topology pdf knots, we should focus on their global “ shape ” of! Μí± ] book boasts of a set 9 8 an arbitrary set, which could consist vectors... A lot of pictures and σ-field structures become quite complex topology emphasizing only the most useful concepts concrete... An arbitrary set, which could consist of vectors metric space topology pdf Rn, functions sequences! Cartesian product of two sets that was studied in MAT108 functions between metric spaces are of! Except where noted otherwise which could consist of vectors in Rn, functions sequences... Discuss probability theory of random processes, the book boasts of a set is defined theorem! A more generalized notion of distance line, in which some of this course is then to metric. And start from scratch then α∈A O α∈C, topology, and Uniform Topologies 18 11 µí± ) is of! Euclidean space, Hausdor spaces, topology, and Uniform Topologies 18 11 has a topology is. Pseudometric space have their corresponding parts in metric space, let x2X, and they are all same. Functions on Un an open set. space hand Written note an arbitrary set, which to!

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